Studies and what each added

The central question is whether recorded aircraft motion fits a globe or the specified flat disc. An IMU (inertial measurement unit) contains gyroscopes and accelerometers. The gyroscope record is the central turning measurement; GPS supplies route and timing, with receiver positions also constraining the fitted track.

This page follows the evidence and the problems each study addressed. Related software checks are grouped together. Current verified findings give the published totals; all 53 stretches and the comprehensive CSV give the individual records. Further planned work is separate.

Establishing usable observations

Study	What it established or added
Reference-file inspection	Established intact packet framing, measurement timing and embedded GPS while preserving the initially unknown IMU payload.
Physical decoding	Independently recovered the reference IMU8 angle and velocity increments, units, axis order and signs.
Timestamp and configuration follow-up	Resolved time conventions and recovered candidate flight windows without choosing them by model agreement.
Other instrument configurations and cross-date IMU6 checks	Tested conversion hypotheses separately instead of applying one instrument's scale to the whole archive.
Installation and clock investigation	Explained changed axis mappings and how fluctuating timestamp intervals can create apparent vertical-force changes.
Screening difficult recordings and remaining motion cases	Recorded keep, reject and unsupported outcomes using data quality and motion criteria; the retained corpus contains 232 originals.
Receiver-position extraction and GPS checks	Built observation streams from raw IMU increments and receiver output while keeping the fused navigation answer out of the fits.
Fixed eligibility inventory	Saved all 53 qualifying stretches before fitting: at least four minutes, every supported speed at least 700 km/h, plus completeness and motion checks.

What the model comparisons resolved

“Resolved” here means that every required model, processing case, repeat and control passed its numerical checks. It does not mean that every model fit equally well. An unfinished calculation provides no finding in favor of either shape.

Study	What it added	Published outcome at that stage
Initial six-recording fits	Tried the complete observation model on real recordings.	Finite outputs, but no completed comparisons.
Archive-wide exploration	Extended the investigation to retained recordings and exposed numerical and motion-model limitations.	Exploratory fitted/unsupported outcomes, separate from the later fast-level selection.
Six-recording refinement	Compared shape first under four instrument/processing assumptions.	Two complete globe preferences; four recordings unresolved.
First full selected-stretch study	Fitted every preselected stretch under the same two primary processing cases.	12 globe / 0 flat / 41 unresolved.
Actual-position stopping qualification	Tested stopping using fresh physical position changes while preserving the gradient and cost limits.	All 24 trial fits passed; three recoveries qualified for the wider run.
Full numerical recovery	Applied the qualified method with unchanged measurements and physical bounds.	30 / 0 / 23, including those trial recoveries.
Paired-recorder precision trial	Qualified wider arithmetic on overlapping recorder pairs, with controls and repeats.	32 / 0 / 21.
Cost-precision recovery	Recovered nine complete comparisons limited by numerical cost precision.	41 / 0 / 12.
Budget continuation	Tested whether a larger finite allowance resolved three exhausted cases.	No additional complete comparisons.
Remaining precision cases	Applied the precision method to the other declared failure classes.	Three more globe preferences: 44 / 0 / 9.
Matched hybrid-solver trial	Combined full bounded steps with damped steps on difficult cases.	Two more complete globe comparisons: 46 / 0 / 7; wider qualification failed.
Six-target hybrid assessment	Tested the remaining declared targets with their own repeats and passing companions.	One more globe preference: 47 / 0 / 6.

These are stages of one developing analysis, not extra independent datasets. The same stretch is counted once in the combined findings. Simultaneous instruments and shared flights also require grouping before statistical claims.

Checking why the models differ

Study	What it solved, tested or left open
Calibration and fit-quality audit	Found a substantial fitting-error gap on the original twelve resolved stretches, together with frequent calibration parameters at their limits.
Individual-epoch residuals	Exposed time-dependent errors and showed that changing the clock at fixed parameters particularly affects vertical predictions.
Refitted clock comparison and calibration audit	All six refitted comparisons completed and preserved globe/field preferences, while calibration readjustment compensated for the clock change.
Z-axis calibration ablation	Fixing Z gain at zero preserved the preference; fixing Z offset left a required fit unfinished.
Joint gain–offset grid	All 27 fits completed and globe/field preferences persisted across the declared Z calibration grid.
Possible onboard field-removal grid	All 27 fits completed and the preferences persisted when a common removed field fraction was allowed.
Second-recording calibration grid and combined audit	Seven complete grid comparisons favored globe; two remained unfinished, preserving the replication's limits.
Within-interval integration trial	Tested an alternative treatment of motion within each IMU interval; four of six fits completed and no replacement qualified.
Reserved-time prediction trial	Tested observations outside the fitting portion; five of six fits completed, leaving the combined comparison unfinished.
Residual covariance audit	Found persistent error correlation, showing why small fitting errors alone do not calibrate confidence.
Training/prediction covariance trial	Tested a training-selected error model on reserved observations; ten of twelve fits completed and the combined comparison remained unresolved.
Shared-flight diagnosis	Compared simultaneous recorder tracks and separated agreement in observations from disagreement in numerical stopping.

The fitting-error audit found median horizontal errors of about 0.068 m for the globe with the specified background field, 29.5 m for zero-field globe and 47.1 m for the disc on the same original twelve stretches and two processing cases. These are in-sample errors under each candidate's declared geometry, not independent prediction accuracy.

Finding and testing numerical failure mechanisms

Software controls have known answers. They check the calculations; they do not add flight evidence. A method that passes them must still be checked on the recordings.

Study or related checks	What it added
Initial solver trial and known-answer qualification	Established software controls, but the observed replacement completed fewer fits than the baseline and did not qualify.
Parameter scaling and verified stopping	Tested alternative numerical handling without changing physical models; neither supplied a qualified general replacement.
Continuation trial and diagnosis	Recovered individual fits but no complete test stretch, motivating a different bounded-step method.
Rectangular-method trial and saved-tangent diagnosis	Exposed sensitivity to starting near parameter limits; passing local linear problems did not establish observed convergence.
Boundary-aware trial and failure diagnosis	Seventeen of 24 fits passed, but prediction-stability failures prevented complete stretch recovery.
Fixed-endpoint probes and precision comparison	Isolated arithmetic/derivative behavior in weak calibration directions before the successful precision recovery.
First damped-solver qualification, diagnosis and revised qualification	Rejected a candidate that failed known optima, then qualified its revised step strategy on 48 controls.
Saved stability audit and native timing probes	Found a timing-boundary discrepancy: a smooth predicted improvement could worsen the actual integrated result.
Packet-aware solver trial	Both controls passed, but its target primary and repeat still failed the original stability gate.
All-six stopping audit	Found timing-boundary behavior in every unfinished primary and reproduced an old-gate rejection at a known cusp minimum.
Two-sided rule, adapter and compiled-motion checks	Passed 43, 30 and two software controls respectively, qualifying a bounded diagnostic rather than changing old labels.
All-six timing-boundary diagnostic	Tests all nineteen unfinished endpoints with two controls; its results need full companion and repeat checks before any stretch can be relabeled.

Supporting controls and earlier simulated-flight work

Work	What it contributed
Geographic, latitude and course controls	Checked model geometry and demonstrated calibration ambiguity even when reconstructed tracks were very accurate.
Longer-interval, coupled-motion and increment-correction controls	Tested how duration, aircraft motion and integration corrections affect recovery of known parameters.
Correlated-error, training-only and nonlinear uncertainty controls	Checked uncertainty calculations while identifying effects that remain uncalibrated for the archive.
Native uncertainty adapter	Checked all 27 parameter columns on six software paths without treating them as observed-data confidence validation.
Synthetic development pilot and matched airline pilot	Tested selection, nuisance effects and decision availability before archival data became the primary project.
Extended airline pilot and observed-route replay	Added longer simulated checks and real route geometry; these remain development simulations, not measured IMU evidence.
Wind, magnetic ambiguity and flight-domain checks	Developed more realistic moving-aircraft nuisance and eligibility models for the former passenger-flight design.

What comes next covers unfinished comparisons, measurement-channel checks, reserved-observation prediction, freely estimating the rotation field and the proposed solar-versus-sidereal rate test. The ground experiment is a separate consumer-IMU project under development.